If my door subsystem is broken, and in the middle of a large fire, is there any other way to open the doors to flush the fire or am I DOOMED?
When the door subsystem is damaged, you cannot open the doors yourself. Crewmembers can open them by walking through them, although I doubt this is useful in a situation where you're trying to vent a room that is on fire.
Eventually, a fire will burn out the oxygen in a room and put itself out, although depending on the strength of your doors, this may be after it's lit nearby rooms on fire.
Personally, any time where I think I'm going to be in a situation where fire is an issue (ie, the "too near a sun" events) I'll preemptively vent as much of the ship as possible. That way, if a fire does break out, it's more likely to burn out quickly.